# Univ.-Prof. Dr. Philipp Grohs

Faculty of Mathematics and
Research Platform Data Science
University of Vienna
Oskar Morgenstern Platz 1
1090 Wien
Austria

### Research Interests

• Approximation Theory and Computational Harmonic Analysis:
• Analysis of nonlinear subdivision algorithms
• Nonlinear wavelet type transformations
• Wavelets, Shearlets,...
• Signal processing
• Phase Retrieval
• Machine Learning with (Convolutional) Neural Networks
• Numerical Methods for PDEs:
• Geodesic finite elements for geometric PDEs (harmonic maps into Riemannian manifolds, ...)
• Numerical solution of kinetic transport PDEs (radiative Transfer, Boltzmann equation, ...)
• Design of optimally convergent adaptive algorithms for PDEs whose solutions possess anisotropic singularities (for instance transport problems or diffusion problems in inhomogeneous media)
• Geometry:
• Discrete differential geometry with applications to architectural geometry
• Lie sphere geometry

### Press

• ETH News press release related to Latsis prize

### Funding

• Intrinsic Approximation of Geometric Data funded by SNF Project 140635
• Sparse Discretization of Kinetic Transport Problems on High-dimensional Phase Spaces (jointly with R. Hiptmair) funded by SNF Project 146356
• Stable Infinite Dimensional Phase Retrieval funded by FWF Project P 30148
• Construction of New Smoothness Spaces on Domains (jointly with H. Feichtinger and S. Dahlke) funded by FWF Project I 3403

### Book

1. Harmonic and Applied Analysis - From Groups to Signals. (Joint work with S. Dahlke, F. DeMari and D. Labate).

### Scientific Publications

1. Smoothness properties of Lie group subdivision schemes. (Joint work with J. Wallner and E. Nava Yazdani).
SIAM Journal on Multiscale Modeling and Simulation 6 (2007), 493-505.
[MR] [doi] [preprint].
2. Log-exponential analogues of univariate subdivision schemes in Lie groups and their smoothness properties. (Joint work with J. Wallner).
In M. Neamtu and L. L. Schumaker, editors, Approximation Theory XII: San Antonio 2007, pages 181-190. Nashboro Press, 2008, ISBN 978-0-9728482-9-9.
[MR] [preprint].
3. Smoothness analysis of subdivision schemes on regular grids by proximity.
SIAM Journal on Numerical Analysis 46 (2008), 2169-2182.
[MR] [doi] [preprint].
4. Smoothness equivalence properties of univariate subdivision schemes and their projection analogues.
Numerische Mathematik 113/2 (2009), 163-180.
[MR] [doi] [preprint] .
5. Smoothness of interpolatory multivariate subdivision in Lie groups.
IMA Journal of Numerical Analysis 29/3 (2009), 760-772.
[MR] [doi] [preprint] .
6. Interpolatory wavelets for manifold-valued data. (Joint work with J. Wallner).
Applied and Computational Harmonic Analysis 27 (2009), 325-333.
[MR] [doi] [preprint] .
7. Laguerre minimal surfaces, isotropic geometry and linear elasticity. (Joint work with H. Pottmann and N. Mitra).
Advances in Computational Mathematics 31/2 (2009), 391-419.
[MR] [doi] [preprint].
8. Approximation order of interpolatory nonlinear subdivision schemes. (Joint work with N. Dyn and J. Wallner).
Journal of Computational and Applied Mathematics 223 (2010), 1697-1703.
[MR] [doi] [preprint].
9. A general proximity analysis of nonlinear subdivision schemes.
SIAM Journal on Mathematical Analysis 42/2 (2010), 729-750.
[MR] [doi] [preprint].
10. Approximation order from stability of nonlinear subdivision schemes.
Journal of Approximation Theory 162 (2010), 1085-1094.
[MR] [doi] [preprint].
11. Edge offset meshes in Laguerre geometry. (Joint work with H. Pottmann and B. Blaschitz).
Advances in Computational Mathematics 33/1 (2010), 45-73.
[MR] [doi] [preprint].
12. Stability of manifold-valued subdivision schemes and multiscale transformations.
Constructive Approximation 32/3 (2010), 569-596.
[MR] [doi] [preprint].
13. The continuous Shearlet Transform: Representation Formulae and Microlocal Analysis.
Oberwolfach Report 7/4 (2010).
[doi]
14. Continuous shearlet tight frames.
Journal of Fourier Analysis and Applications 17/3 (2011), 506-518.
[doi] [arXiv] [preprint].
15. Continuous Shearlet frames and Resolution of the Wavefront Set.
Monatshefte für Mathematik 164/4 (2011), 393-426.
[doi] [arXiv] [preprint].
16. Interpolating composite systems.
In M. Neamtu and L. L. Schumaker, editors, Approximation Theory XIII: San Antonio 2010, Springer Proceedings in Mathematics 13 (2012), 111-120.
[doi] [preprint].
17. Shearlets and Microlocal Analysis.
In: Shearlets: Multiscale Analysis for Multivariate Data, Birkhäuser-Springer (2012).
[doi] [book] [preprint].
18. Ridgelet-type Frame Decompositions for Sobolev Spaces related to linear Transport.
Journal of Fourier Analysis and Applications 18/2 (2012), 309-325.
[doi] [preprint].
19. Definability and stability of multiscale decompositions for manifold-valued data. (Joint work with J. Wallner).
Journal of the Franklin Institute 349/5 (2012), 1648-1664.
[doi] [arXiv] [preprint].
20. Tree Approximation with anisotropic decompositions.
Applied and Computational Harmonic Analysis 33 (2012), 44-57.
[doi] [preprint].
21. Tree Approximation and optimal Image Coding with Shearlets.
Proceedings SampTA: Singapore 2011 (2012).
[preprint].
22. On the Structure of Anisotropic Frames.
Oberwolfach Report 29 (2012).
[doi]
23. Ruled Laguerre minimal surfaces. (Joint work with H. Pottmann and M. Skopenkov).
Mathematische Zeitschrift 272/1 (2012), 645-674.
[doi] [arXiv] [preprint].
24. Bandlimited Shearlet Frames with nice Duals.
Journal of Computational and Applied Mathematics 243 (2013), 139-151.
[doi] [preprint].
25. Refinable functions for dilation families.
Advances in Computational Mathematics 38/3 (2013), 531-561.
[doi] [arXiv] [preprint].
26. Intrinsic Localization of Anisotropic Frames.
Applied and Computational Harmonic Analysis 35/2 (2013), 264-283.
[doi] [preprint].
27. Finite Elements of Arbitrary Order and Quasiinterpolation for Riemannian Data.
IMA Journal of Numerical Analysis 33/3 (2013), 849-874.
[doi] [preprint].
28. Geometric Multiscale Decompositions of Dynamic Low-Rank Matrices.
Computer Aided Geometric Design 30 (2013), 805-826.
[doi] [preprint].
29. Parabolic Molecules. (Joint work with G. Kutyniok).
Foundations of Computational Mathematics 14/2 (2014), 299-337.
[doi] [preprint].
30. Parabolic Molecules: Curvelets, Shearlets, and Beyond. (Joint work with S. Keiper, G. Kutyniok, M. Schaefer).
In Approximation Theory XIV: San Antonio 2013, Springer Proceedings in Mathematics (2014), to appear.
[preprint].
31. $\alpha$-Molecules: Curvelets, Shearlets, Ridgelets, and Beyond. (Joint work with S. Keiper, G. Kutyniok, M. Schaefer).
In Wavelets and Sparsity XV (San Diego, CA, 2013), SPIE Proc. 8858, SPIE, Bellingham, WA, (2014), to appear.
[preprint].
32. Wolfowitz's Theorem and Convergence of Consensus Algorithms in Hadamard Spaces.
Proceedings of the American Mathematical Society (2014), to appear.
[preprint].
33. Hyperbolic Cross Approximation for the Spatially Homogenous Boltzmann Equation. (Joint work with E. Fonn and R. Hiptmair).
IMA Journal of Numerical Analysis (2014), to appear.
[doi] [preprint].
34. Intrinsic Localization of Anisotropic Frames II: $\alpha$-molecules (Joint work with S. Vigogna).
Journal of Fourier Analysis and Applications (2014), to appear.
[doi] [preprint].
35. Cartoon Approximation with $\alpha$-Curvelets (Joint work with S. Keiper, G. Kutyniok and M. Schaefer).
Journal of Fourier Analysis and Applications (2014), to appear.
[preprint].
36. Optimal A Priori Discretization Error Bounds for Geodesic Finite Elements. (Joint work with H. Hardering and O. Sander).
Foundations of Computational Mathematics (2014), to appear.
[doi] [preprint].
37. FFRT-A Fast Finite Ridgelet Transform for Radiative Transport (Joint work with S. Etter and A. Obermeier).
SIAM Journal on Multiscale Modeling and Simulation (2014), to appear.
[preprint] [MATLAB codes].
38. $\varepsilon$-Subgradient Algorithms for Locally Lipschitz Functions on Riemannian Manifolds (Joint work with S. Hosseini).
Advances in Computational Mathematics (2015), to appear.
[preprint].
39. Optimal Adaptive Ridgelet Schemes for Linear Transport Equations (Joint work with A. Obermeier).
Applied and Computational Harmonic Analysis (2015), to appear.
[preprint].
40. Ridgelet Methods for Linear Transport Equations (Joint work with A. Obermeier).
In Curves and Surfaces: Paris 2014, Springer Proceedings in Mathematics (2015), to appear.
[preprint]
41. Nonsmooth Trust Region Algorithms for Locally Lipschitz Functions on Riemannian Manifolds (Joint work with S. Hosseini).
IMA Journal of Numerical Analysis (2015), to appear.
[preprint].
42. $\alpha$-Molecules (Joint work with S. Keiper, G. Kutyniok and M. Schaefer).
Applied and Computational Harmonic Analysis (2015), to appear.
[preprint].
43. Total Variation Regularization by Iteratively Reweighted Least Squares on Hadamard Spaces and the Sphere (Joint work with M. Sprecher).
IMA Journal on Information and Inference (2015), to appear.
[preprint]
44. Deep convolutional neural networks on cartoon functions (Joint work with T. Wiatowski, and H. Boelcskei).
Proc. of IEEE International Symposium on Information Theory (ISIT), Barcelona, Spain, (2016), to appear.
[preprint]
45. Discrete Deep Feature Extraction: A Theory and New Architectures (Joint work with T. Wiatowski, M. Tschannen, A. Stanic and H. Boelcskei).
Journal of Machine Learning Research (Proc. ICML 2016) (2016), to appear.
[preprint]
46. A Shearlet-Based Fast Thresholded Landweber Algorithm for Deconvolution (Joint work with Z. Kereta and U. Wiesmann).
International Journal of Wavelets, Multiresolution and Information Processing (2016), to appear.
[preprint].
47. Scattered Manifold-Valued Data Approximation (Joint work with M. Sprecher and T. P.-Y. Yu).
Numerische Mathematik (2016), to appear.
[preprint].
48. Reconstructing real-valued functions from unsigned coefficients with respect to wavelet and other frames (Joint work with R. Al-Aifari, I. Daubechies and G. Thakur).
Journal of Fourier Analysis and Applications (2017), to appear.
[preprint]
49. Phase Retrieval in the general Setting of Continuous Banach Frames (Joint work with R. Al-Aifari).
SIAM Journal on Mathematical Analysis (2017), to appear.
[preprint]
50. On the Approximation of Functions with Line Singularities by Ridgelets (Joint work with A. Obermeier).
Journal of Approximation Theory (2017), to appear.
[preprint]
51. Energy decay and conservation in deep convolutional neural networks (Joint work with H. Boelcskei and T. Wiatowski).
Proc. of IEEE International Symposium on Information Theory (ISIT), Aachen, Germany, (2017), to appear.
[preprint]
52. Tensor-product discretization for the spatially inhomogeneous and transient Boltzmann equation in 2D (Joint work with R. Hiptmair and S. Pintarelli).
SMAI Journal of Computational Mathematics (2017), to appear.
[preprint]
53. Topology Reduction in Deep Convolutional Neural Networks (Joint work with H. Boelcskei and T. Wiatowski).
Proc. of SPIE (Wavelets and Sparsity XVII), San Diego, USA, July 2017, to appear, (invited paper)
[preprint]
54. Memory Optimal Neural Network Approximation (Joint work with H. Boelcskei, G. Kutyniok and P. Petersen).
Proc. of SPIE (Wavelets and Sparsity XVII), San Diego, USA, July 2017, to appear, (invited paper)
[preprint]
55. Edge and Corner Detection with Parabolic Dictionaries (Joint work with Z. Kereta).
Applied and Computational Harmonic Analysis (2018).
[preprint]
56. Energy Propagation in Deep Convolutional Neural Networks (Joint work with H. Boelcskei and T. Wiatowski).
IEEE Transactions on Information Theory (2018), to appear.
[preprint]
57. Solving linear Kolmogorov Equations by means of Deep Learning.
Oberwolfach Report 14 (2018)
[doi]
58. Stable Phase Retrieval in Infinite Dimensions (Joint work with R. Al-Aifari, I. Daubechies and R. Yin).
Foundations of Computational Mathematics (2018), to appear.
[preprint]
59. Stable Gabor Phase Retrieval and Spectral Clustering (Joint work with M. Rathmair).
Communications on Pure and Applied Mathematics (2018), to appear.
[preprint]
60. Optimal Approximation with Sparsely Connected Deep Neural Networks (Joint work with H. Boelcskei, G. Kutyniok and P. Petersen).
SIAM Journal on Mathematics of Data Science (2019), to appear.
[preprint]
61. Projection-Based Finite Elements for Nonlinear Function Spaces (Joint work with H. Hardering, O. Sander and M. Sprecher).
SIAM Journal on Numerical Analysis (2019), to appear.
[preprint]
62. Towards a regularity theory for ReLU networks -- chain rule and global error estimates (joint work with J. Berner, D. Elbraechter and A. Jentzen).
Proceedings SampTA: Bordeaux 2019 (2019).
[preprint]
63. A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations (joint work with F. Hornung, A. Jentzen and P. Von Wurstemberger).
Memoirs of the American Mathematical Society (2019), to appear.
[preprint]
64. Gabor Phase Retrieval is Severely Ill-Posed (Joint work with R. Alaifari).
Applied and Computational Harmonic Analysis (2019), to appear.
[preprint]
65. Analysis of the generalization error: Empirical risk minimization over deep artificial neural networks overcomes the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations (joint work with J. Berner and A. Jentzen).
SIAM Journal on Mathematics of Data Science (2019), to appear.
[preprint]
66. How degenerate is the parametrization of neural networks with the ReLU activation function? (joint work with J. Berner and D. Elbraechter). Proceedings of NeurIPS 2019, to appear.
[preprint]
67. Planting Synchronisation Trees for Discovering Interaction Patterns among Brain Regions (joint work with L. Bauer, C. Plant and A. Wohlschlaeger).
IEEE International Conference on Data Mining 2019: PhD Forum, in press.
68. Phase Retrieval: Uniqueness and Stability (joint work with S. Koppensteiner and M. Rathmair).
SIAM Review, to appear.
[preprint]
69. On the maxima of cardinal refinable functions with positive symbol.
Geometry preprint 2009/03, TU Graz, August 2009.
[preprint].
70. Sparse Twisted Tensor Frame Discretizations of Parametric Transport Equations. (Joint work with Ch. Schwab).
SAM Report 2011-41, ETH Zurich, May 2011.
[preprint].
71. Projection-based Quasiinterpolation in Manifolds. (Joint work with M. Sprecher).
SAM Report 2013-23, ETH Zurich, July 2013.
[preprint] [MATLAB codes].
72. Polar Spectral Scheme for the Spatially Homogenous Boltzmann Equation (Joint work with E. Fonn and R. Hiptmair).
SAM Report 2014-13, ETH Zurich, April 2014.
[preprint].
73. Continuous Parabolic Molecules (Joint work with Z. Kereta).
SAM Report 2015-17, ETH Zurich, July 2015.
[preprint]
74. Anisotropic Multiscale Systems on Bounded Domains (Joint work with G. Kutyniok, J. Ma and P. Petersen).
SAM Report 2015-30, ETH Zurich, September 2015.
[preprint]
75. Edge and Corner Detection with Parabolic Dictionaries (Joint work with Z. Kereta).
SAM Report 2016-40, ETH Zurich, September 2016.
[preprint]
76. A uniform bound on the growth of the norm of the logarithmic derivative on certain growth classes of entire functions and applications in Phase Retrieval (Joint work with M. Rathmair).
April 2018.
[preprint]
77. Solving stochastic differential equations and Kolmogorov equations by means of deep learning (Joint work with C. Beck, S. Becker, N. Jaafari and A. Jentzen).
May 2018
[preprint]
78. The Universal Approximation Power of Finite-Width Deep ReLU Networks (joint work with H. Boelcskei, D. Elbraechter and D. Prerekrestenko).
June 2018
[preprint]
79. DNN Expression Rate Analysis of High-dimensional PDEs: Application to Option Pricing (joint work with D. Elbraechter, A. Jentzen and C. Schwab).
September 2018
[preprint]
80. Deep Neural Network Approximation Theory (joint work with H. Boelcskei, D. Elbraechter and D. Perekrestenko).
January 2019
[preprint]
81. The Oracle of DLPhi (joint work with the participants of the Oberwolfach Seminar: Mathematics of Deep Learning -- this manuscript has mostly humouristic value. However, while the main theorem is based on a dubious application of the axiom of choice, it is a correct mathematical statement. Therefore, this manuscript at least highlights the dangers of applying mathematical theory to real-world applications blindly).
January 2019
[pdf]
82. Stable Gabor phase retrieval for multivariate functions (joint work with M. Rathmair).
March 2019
[preprint]
83. Space-time error estimates for deep neural network approximations for differential equations (joint work with F. Hornung, A. Jentzen and P. Zimmermann).
September 2019
[preprint]
84. Deep neural network approximations for Monte Carlo algorithms (joint work with A. Jentzen and D. Salimova).
September 2019
[preprint]
85. Uniform error estimates for artificial neural network approximations for heat equations (joint work with L. Gonon, A. Jentzen, D. Kofler, D. Siska).
December 2019
[preprint]
86. Group testing for SARS-CoV-2 allows for up to 10-fold efficiency increase acrossrealistic scenarios and testing strategies (joint work with C.M. Verdun, T. Fuchs, P. Harar, D. Elbraechter, D. Fischer, J. Berner, F. Theis and F. Krahmer).
May 2020
[preprint] [web-app]
87. Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditions (joint work with L. Herrmann).
July 2020
[preprint]

### Theses

1. Approximation Theory in Manifolds.
Habilitation Thesis, TU Graz, 2010.
[full thesis] [introduction].
2. Smoothness Analysis of nonlinear Subdivison Schemes on regular grids.
Phd Thesis, TU-Wien, 2007.
[pdf].
3. Asymptotik von Goppa Codes (in German).
Diploma Thesis, TU-Wien, 2006.
[pdf].
4. Beweis der Bieberbachschen Vermutung (in German).
Semester Thesis, TU-Wien, 2005.
[pdf].